1) This picture basically explains it all, cold air is more dense because with warm air the gas particles will expand more (higher volume). Using the density=mass/volume this means that density and volume are inversely proportional. So an increase in volume (warm air) will cause a decrease in density and therefore be less dense. However the warm air particles are moving more because an increase in temperature will cause the molecules to move around more (more kinetic energy).
2) The amount of water vapor in the air effects the density of the air. Water vapor is a light gas when it is compared to Oxygen and Nitrogen, this means that when water vapor increases the amount of of oxygen and nitrogen will decrease. This causes the density to decrease. This means that moist air will be less dense than dry air because it has more water vapor.